On a magnetic characterization of spectral minimal partitions

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Given a bounded open set Ω in ℝn (or in a Riemannian manifold) and a partition of Ω by k open sets Dj, we consider the quantity 𝚖𝚊𝚡jλ(Dj) where λ(Dj) is the ground state energy of the Dirichlet realization of the Laplacian in Dj. If we denote by ℒk(Ω) the infimum over all the k-partitions of 𝚖𝚊𝚡jλ(Dj), a minimal k-partition is then a partition which realizes the infimum. When k=2, we find the two nodal domains of a second eigenfunction, but the analysis of higher k’s is non trivial and quite interesting. In this paper, we give the proof of one conjecture formulated in [5] and [16] about a magnetic characterization of the minimal partitions when n=2.

@article{Helffer2013, abstract = {Given a bounded open set $\Omega $ in $\mathbb \{R\}^n$ (or in a Riemannian manifold) and a partition of $\Omega $ by $k$ open sets $D_j$, we consider the quantity $\texttt \{max\}_j\lambda (D_j)$ where $\lambda (D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $\mathcal \{L\}_k(\Omega )$ the infimum over all the $k$-partitions of $\texttt \{max\}_j\lambda (D_j)$, a minimal $k$-partition is then a partition which realizes the infimum. When $k=2$, we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$’s is non trivial and quite interesting. In this paper, we give the proof of one conjecture formulated in [5] and [16] about a magnetic characterization of the minimal partitions when $n=2$.}, author = {Helffer, Bernard, Hoffmann-Ostenhof, Thomas}, journal = {Journal of the European Mathematical Society}, keywords = {minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theorem; minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theorem}, language = {eng}, number = {6}, pages = {2081-2092}, publisher = {European Mathematical Society Publishing House}, title = {On a magnetic characterization of spectral minimal partitions}, url = {http://eudml.org/doc/277728}, volume = {015}, year = {2013},}

TY - JOURAU - Helffer, BernardAU - Hoffmann-Ostenhof, ThomasTI - On a magnetic characterization of spectral minimal partitionsJO - Journal of the European Mathematical SocietyPY - 2013PB - European Mathematical Society Publishing HouseVL - 015IS - 6SP - 2081EP - 2092AB - Given a bounded open set $\Omega $ in $\mathbb {R}^n$ (or in a Riemannian manifold) and a partition of $\Omega $ by $k$ open sets $D_j$, we consider the quantity $\texttt {max}_j\lambda (D_j)$ where $\lambda (D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $\mathcal {L}_k(\Omega )$ the infimum over all the $k$-partitions of $\texttt {max}_j\lambda (D_j)$, a minimal $k$-partition is then a partition which realizes the infimum. When $k=2$, we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$’s is non trivial and quite interesting. In this paper, we give the proof of one conjecture formulated in [5] and [16] about a magnetic characterization of the minimal partitions when $n=2$.LA - engKW - minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theorem; minimal partitions; nodal sets; Aharonov-Bohm Hamiltonians; Courant's nodal theoremUR - http://eudml.org/doc/277728ER -